Phase retrieval
Important applications of phase retrieval include X-ray crystallography, transmission electron microscopy and coherent diffractive imaging, for which
[1] Uniqueness theorems for both 1-D and 2-D cases of the phase retrieval problem, including the phaseless 1-D inverse scattering problem, were proven by Klibanov and his collaborators (see References).
Here we consider 1-D discrete Fourier transform (DFT) phase retrieval problem.
Since the DFT operator is bijective, this is equivalent to recovering the phase
It is common recovering a signal from its autocorrelation sequence instead of its Fourier magnitude.
, calculated from the diffraction pattern via the signal equation[clarification needed], is then substituted for
, giving an estimate of the Fourier transform: where the ' denotes an intermediate result that will be discarded later on.
, satisfies the object constraints[clarification needed].
Theoretically, the process will always lead to a convergence,[1] but the large number of iterations needed to produce a satisfactory image (generally >2000) results in the error-reduction algorithm by itself being unsuitable for practical applications.
The chief advantage of the hybrid input-output algorithm is that the function
contains feedback information concerning previous iterations, reducing the probability of stagnation.
Its convergence rate can be further improved through step size optimization algorithms.
{Scientific Reports volume 8, Article number: 6436 (2018)} For a two dimensional phase retrieval problem, there is a degeneracy of solutions as
This leads to "image twinning" in which the phase retrieval algorithm stagnates producing an image with features of both the object and its conjugate.
[3] The shrinkwrap technique periodically updates the estimate of the support by low-pass filtering the current estimate of the object amplitude (by convolution with a Gaussian) and applying a threshold, leading to a reduction in the image ambiguity.
To uniquely identify the underlying signal, in addition to the methods that adds additional prior information like Gerchberg–Saxton algorithm, the other way is to add magnitude-only measurements like short time Fourier transform (STFT).
The method introduced below mainly based on the work of Jaganathan et al.[5] Given a discrete signal
denotes the separation in time between adjacent short-time sections and the parameter
denotes the number of short-time sections considered.
In fact, however, for the most cases in practical we only need to consider the measurements corresponding to
can be uniquely identified from its STFT magnitude if the following requirements are satisfied: The proof can be found in Jaganathan' s work,[5] which reformulates STFT phase retrieval as the following least-squares problem:
The algorithm, although without theoretical recovery guarantees, empirically able to converge to the global minimum when there is substantial overlap between adjacent short-time sections.
To establish recovery guarantees, one way is to formulate the problems as a semidefinite program (SDP), by embedding the problem in a higher dimensional space using the transformation
and relax the rank-one constraint to obtain a convex program.
Phase retrieval is a key component of coherent diffraction imaging (CDI).
In CDI, the intensity of the diffraction pattern scattered from a target is measured.
In this way, phase retrieval allows for the conversion of a diffraction pattern into an image without an optical lens.
Using phase retrieval algorithms, it is possible to characterize complex optical systems and their aberrations.
[6] For example, phase retrieval was used to diagnose and repair the flawed optics of the Hubble Space Telescope.
[7][8] Other applications of phase retrieval include X-ray crystallography[9] and transmission electron microscopy.
